Hang rest of paragraph after specific point (or token)

Say I have some point in a sentence that I want the rest of the paragraph to hang after; result like this:

PART / Long title that won't fit on one line that
we want hanged after the slash on the next line

I made this work ish by calculating the width of a box containing the PART / prefix, like so:

\documentclass{article}
% Prevent extra indent
\setlength\parindent{0pt}
% Box to contain the stuff that the text will hang after
\newbox\partprefixBox
% The stuff that the text will hang after
\def\partprefix{\Large PART /\nobreakspace}
% Set part-header-prefix into the box
\setbox\partprefixBox=\hbox{\partprefix}
\begin{document}
\bgroup
% Set the hangindent size (using the box width)
\hangindent=\wd\partprefixBox%
\hangafter=1%
\partprefix Long title that won't fit on one line that we want hanged after the slash on the next line\par
\egroup
\end{document}

I'm wondering if there is a more robust way of doing this, and whether there is a way to do it with centered text (i.e. if the text is not gonna break, then it should be centered).

Ideally something like a \hanghere command that would work like so:

PART / \hanghere Long title that won't fit on one line that we want hanged after the slash on the next line

or this syntax would also work

\hanghere{PART / }Long title that won't fit on one line that we want hanged after the slash on the next line